1 Each other and one another mean the same.

Mary and I write to each other/one another every day.

They sat without looking at each other/one another.

2 There is a possessive each other’s/one another’s.

We often borrow each other’s clothes.

They stood looking into one another’s eyes.

Each other/one another are not used as subjects.

We must each listen care fully to what the other says.

(NOT… We must listen carefully to what each other say.)

3 Note the difference between each other/one another and ourselves /yourselves / themselves. Compare:

They were looking at each other. A ← B each other
→

(= Each person was looking at the other.)

They were looking at themselves. A → A themselves

B ← B

(= Each person was looking at him- or herself.)